To determine the domain, range, and asymptote of the function [tex]\( h(x) = 2^{x + 4} \)[/tex], let's examine each aspect of the function in detail.
### Domain
The domain of a function refers to all the input values (x-values) for which the function is defined. For exponential functions of the form [tex]\( a^{bx + c} \)[/tex] (where [tex]\( a > 0 \)[/tex] and [tex]\( a \neq 1 \)[/tex]), the exponent can be any real number. Therefore, [tex]\( h(x) = 2^{x + 4} \)[/tex] is defined for all real numbers [tex]\( x \)[/tex].
Hence, the domain is:
[tex]\[ \{ x \mid x \text{ is a real number} \} \][/tex]
### Range
The range of a function refers to all the possible output values (y-values) the function can produce. For the exponential function [tex]\( h(x) = 2^{x + 4} \)[/tex], the base [tex]\( 2 \)[/tex] raised to any real number is always positive. Exponential functions of this form (with a positive base greater than 1) will never produce zero or negative values.
Thus, the range is:
[tex]\[ \{ y \mid y > 0 \} \][/tex]
### Asymptote
The asymptote of a function refers to a line that the graph of the function approaches but never touches. For the exponential function [tex]\( h(x) = 2^{x + 4} \)[/tex], as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( 2^{x + 4} \)[/tex] approaches zero but never actually reaches zero. This behavior indicates a horizontal asymptote at [tex]\( y = 0 \)[/tex].
Therefore, the asymptote is:
[tex]\[ y = 0 \][/tex]
### Summary
Combining these observations, we have:
- Domain: [tex]\( \{ x \mid x \text{ is a real number} \} \)[/tex]
- Range: [tex]\( \{ y \mid y > 0 \} \)[/tex]
- Asymptote: [tex]\( y = 0 \)[/tex]
Thus the correct option is:
[tex]\[ \{ x \mid x \text{ is a real number} \}; \text{range: } \{ y \mid y > 0 \}; \text{asymptote: } y = 0 \][/tex]
Therefore, the correct option from the given choices is:
[tex]\[ \boxed{\text{domain: } \{ x \mid x \text{ is a real number} \}; \text{range: } \{ y \mid y > 0 \}; \text{asymptote: } y = 0} \][/tex]
